Here's a map of places where you can eat lunch: dining map. Some suggested places: Bytes, Ike's Place (for sandwiches), Coupa Cafe (crepes), Nexus in the James Clar Center (many options).
| June 19 | June 20 | June 21 | June 22 | |
| 9:30-10:30 | Ye Tian | Kartik Prasanna | Kartik Prasanna | Akshay Venkatesh |
| 11-11:45 | Pei-Yu Tsai | Jerry Wang | Luis Garcia | (10:45 -- 11:30) Paul Nelson |
| 11:45-12:30 | Mike Lipnowski | Daniel Disegni | (11:45--12:45) Chris Skinner | |
| 2-3 | Wei Zhang | Wei Zhang | Wei Zhang | |
| 3:30-4:30 | Samit Dasgupta | Yifeng Liu | Dick Gross |
This talk wasn't delivered at the conference but the slides are available: slides
Abstract: Let G be a reductive group over the field k, and V a representation of G over k. We study the fiber of the map V --> V//G over a k-rational point of the canonical quotient, using techniques in Galois cohomology. We then specialize to the case where G = SL(W) and V = Sym^2(W)* + Sym^2(W)*, calculate the G-orbits in the fiber over a non-degenerate binary form in V//G, and describe their relation with 2-coverings of hyperelliptic curves over k. This is joint work with Manjul Bhargava and Jerry Wang.
Abstract: We will discuss the relative trace formula approach to the Gan-Gross-Prasad conjecture and the refinement of Ichino-Ikeda and N. Harris for U(n)xU(n+1).
Abstract: We use both Waldspurger's period formula and Gross-Zagier formula to extend Heegner's result on congruent number problem and obtaine some evidence on 2-adic part of BSD conjecture for congruent elliptic curves.
Abstract: Let k be a p-adic local field and \pi be an irreducible smooth representation of PGL(2,k). The theory of local newforms established by Casselman shows that the fixed vectors of the congruence subgroups \Gamma_0(p^m) of PGL(2,k) encode many invariants of the representation like the L-function and the \epsilon-factor associated it. In this talk, I will define an analogous notion of the newforms for SO(2n+1,k) by considering a family of open compact subgroups that generalizes the groups \Gamma_0(p^m).
Abstract: We'll discuss a p-adic criteria for elliptic curves to have algebraic and analytic rank one, along with various applications.
Abstract: What does the Gross-Zagier formula say about the *full* Birch and Swinnerton-Dyer conjecture? For elliptic curves over Q of analytic rank one, it implies the conjectured formula up to a nonzero rational number. For abelian varieties parametrized by Shimura curves, the analogous application of Zhang's Gross-Zagier formula depends on a conjecture on periods of automorphic forms. I will make some propaganda for this conjecture, then explain how p-adic analogues of the Gross-Zagier formula due to Perrin-Riou, Kobayashi and myself allow to go on and study the BSD formula up to p-integrality.
Abstract:
In the first talk I will discuss two seemingly unrelated problems:
A. A "missing" case of the Gross-Zagier theorem; this is a case where the Rankin-Selberg L-function vanishes for sign reasons but one does not know how to make the associated algebraic cycle(s) predicted by Bloch-Beilinson.
B. Integral period relations between quaternionic modular forms.
In the second talk I will explain an approach to studying Problem B and how this leads to a plausible construction of a cycle in Problem A. Everything above is joint work with Atsushi Ichino.
Abstract: Recently Manjul Bhargava and his collaborators have computed the average sizes of several Selmer groups, thereby obtaining unconditional bounds on the average rank of the corresponding Mordell-Weil group. In this talk, I will explain about the counting technique developed by Bhargava and how it is used to obtain average size of 2-Selmer groups of Jacobians of hyperelliptic curves with a marked non-Weierstrass point. This is joint work with Arul Shankar.
Abstract: The Cheeger-Muller theorem provides a spectral means of obtaining information about torsion in the cohomology of compact manifolds. We discuss its utility for proving comparisons between torsion in the cohomology of two different manifolds of arithmetic origin.
Abstract: Given a cohomology class, say a Hecke eigenclass, for an arithmetic group, we can ask about its lattice of periods. I will discuss some (still speculative) conjectures about what this lattice actually is.
Abstract: This is a report on work in progress with Matthew Greenberg. Let $F$ be a totally real field and let $p$ be a rational prime that for simplicity we assume is inert in $F$. Let $f$ be a modular form for a totally indefinite quaternion algebra $A$ over $F$ whose local representation at $p$ is Steinberg. We define a Darmon $\mathcal{L}$-invariant attached to $f$, which is a vector of $p$-adic numbers indexed by the embeddings of $F$ into $\mathbf{C}_p$. This $\mathcal{L}$-invariant is defined using the cohomology of a $p$-arithmetic subgroup of $A^*$ and is modeled after Darmon's definition of the $L$-invariant in the case $F=\mathbf{Q}, A=\mathrm{M}_2(\mathbf{Q})$. Next we consider certain $p$-adic Banach space representations of $\mathrm{GL}_2(F_p)$ denoted $B(k,\mathcal{L})$, where $k$ is the vector of weights of $f$ and $\mathcal{L}$ is a vector as before. These Banach space representations generalize the construction of Breuil in the case $F=\mathbf{Q}$, and build on work of Schraen. Our main result is that there exist $\mathrm{GL}_2(F_p)$-interwining operators from $B(k,\mathcal{L})$ to the $f$-isotypic component of the completed cohomology of $A$ if and only if the vector $\mathcal{L}$ is the negative of the Darmon $\mathcal{L}$-invariant. This generalizes a result of Breuil in the case $F=\mathbf{Q}, A=\mathrm{M}_2(\mathbf{Q})$.
Abstract: We will define certain anti-cyclotomic p-adic L-functions for some cuspidal automorphic representations of GL_2(F) where F is a totally real field, and relate the special value of such L-function to the p-adic logarithm of Heegner points on the Abelian variety (or more generally, motive) of GL(2)-type attached to the representation. In particular, this provides a new criterion to tell whether these Heegner points are torsion or not. This is a generalization of the recent work of Bertolini, Darmon and Prasanna for which we put the story into the general picture of Waldspurger, Yuan-Zhang-Zhang's generalization of Gross-Zagier formula, and Gan-Gross-Prasad. It is a joint work with Shouwu Zhang and Wei Zhang.
Abstract: Consider two different holomorphic Hecke eigenforms $f_i \in \pi_i$, $i=1,2$ of weight $2$ on a Shimura curve $X/\mathbb{Q}$. We will first discuss Beilinson's conjecture relating the image of the complex regulator map from a higher Chow group with the special value of $L(\pi_1 \times \pi_2,s)$ at $s=0$. Then we will review Borcherds's construction of meromorphic functions on $X$ with divisors supported on CM points. Finally we will show how to use the theta correspondence to compute, assuming that the $f_i$ have full level and up to an archimedean zeta integral, the period integrals arising as regulators of higher Chow cycles constructed using Borcherds' functions.
I will discuss an approach to studying the limiting behavior of automorphic forms on quaternion algebras that succeeds admirably when the levels involved are large squares, and describe some new technical problems that arise in trying to relax this restriction.